How can I thicken this area?
Just to answer the question, disregarding how the geometry looks:
Try to do a soft selection and go on from there.

So first go into edit mode.

Then enable proportional editing. (press the round dot/circle icon at the bottom. Its the one left of the magnet symbol. End select "Connected")
Now you can select a bunch of faces or vertices and use your scrollwheel. The circle that appears marks the soft selection area.

If you drag your vertices outward (using scale or simple move) the neigbours are dragged along with it.

You can influence the shape you are dragging by selection a different shape from the proportional editing menu.